What is the angle between #<5 , 5 , 3 > # and # < 4, 9 , 1 > #?

1 Answer
Jul 19, 2016

#26.58 # degrees

Explanation:

We're going to use the dot product here. For two vectors #vec(u),vec(v) in RR^3# where #vec(u) = (u_1,u_2,u_3) and vec(v) = (v_1,v_2,v_3)# the dot product is given by the following two formulae:

#vec(u)*vec(v) = u_1v_1 + u_2v_2+u_3v_3#

and

#vec(u)*vec(v) = |vec(u)||vec(v)|costheta#

Combining these gives:

#theta = cos^(-1)((u_1v_1 + u_2v_2+u_3v_3)/(|vec(u)||vec(v)|))#

#|vec(u)| = sqrt(5^2+5^2+3^2) = sqrt(59)#

#|vec(v)| = sqrt(4^2+9^2+1^2) = sqrt(98)#

#theta = cos^(-1)((5*4 + 5*9 + 3*1)/(sqrt(59)sqrt(98))) = 26.58 # degrees